It is useful to simply investigate angular momentum with just one free rotation angle. This might be the quantum plane propeller. We will do a good job of this in 3 dimensions later.
Lets assume we have a mass constrained to move in a circle of radius . Assume the motion in the circle is free, so there is no potential. The kinetic energy is for .
If we measure distance around the circle, then
and the one problem we have is that once I go
completely around the circle, I am back to
Lets just go ahead and write our wavefunction.
This must be single valued so we need to require that
This will prove to be true for 3 dimensions too, however, the 3 components of angular momentum do not commute with each other, leading to all kinds of fun.
Jim Branson 2013-04-22